PHYSICAL REVIEW FLUIDS 3, 103302 (2018)

Drag enhancement and drag reduction in viscoelastic flow

Atul Varshney1,2 and Victor Steinberg1,3

1Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel2Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria

3Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

(Received 29 May 2018; published 15 October 2018)

Creeping flow of polymeric fluid without inertia exhibits elastic instabilities and elasticturbulence accompanied by drag enhancement due to elastic stress produced by flow-stretched polymers. However, in inertia-dominated flow at high Re and low fluid elasticityEl, a reduction in turbulent frictional drag is caused by an intricate competition betweeninertial and elastic stresses. Here we explore the effect of inertia on the stability ofviscoelastic flow in a broad range of control parameters El and (Re, Wi). We presentthe stability diagram of observed flow regimes in Wi-Re coordinates and find that theinstabilities’ onsets show an unexpectedly nonmonotonic dependence on El. Further, threedistinct regions in the diagram are identified based on El. Strikingly, for high-elasticityfluids we discover a complete relaminarization of flow at Reynolds number in the rangeof 1 to 10, different from a well-known turbulent drag reduction. These counterintuitiveeffects may be explained by a finite polymer extensibility and a suppression of vorticity athigh Wi. Our results call for further theoretical and numerical development to uncover therole of inertial effect on elastic turbulence in a viscoelastic flow.

DOI: 10.1103/PhysRevFluids.3.103302

I. INTRODUCTION

Long-chain polymer molecules in Newtonian fluid alter the rheological properties of the fluid;the relation between stress and strain becomes nonlinear. Moreover, polymers being stretched bya velocity gradient in shear flow engender elastic stress that modifies the flow via a feedbackmechanism. It results in pure elastic instabilities [1,2] and elastic turbulence (ET) [3], observedat Re � 1 and Wi � 1. Here Re is the ratio of inertial to viscous stresses, Re = UDρ/η, andWi defines the degree of polymer stretching Wi = λU/D, where U is the flow speed, D is thecharacteristic length scale, λ is the longest polymer relaxation time, and ρ and η are the density anddynamic viscosity of the fluid, respectively [4].

Elastic turbulence is a spatially smooth, random-in-time chaotic flow, whose statistical, mean,and spectral properties are characterized experimentally [3,5–11], theoretically [12,13], and numer-ically [14–17]. The hallmark of ET is a steep power-law decay of the velocity power spectrum withan exponent |α| > 3 indicating that only a few modes are relevant to flow dynamics [3,5,12,13].Further, an injection of polymers into a turbulent flow of Newtonian fluid at Re � 1 reduces thedrag and also has a dramatic effect on the turbulent flow structures [18]. In recent investigations, adifferent state of small-scale turbulence associated with maximum drag reduction asymptotes wasobserved in a pipe flow at Re � 1 and Wi � 1. This state is termed elasto-inertial turbulence (EIT)and exhibits properties similar to ET despite the fact that it is driven by both inertial and elasticstresses and their interplay defines EIT properties [19–21]. Thus, the fundamental question ariseshow the inertial effect modifies ET in viscoelastic flow towards turbulent drag reduction.

Numerous studies were performed in various flow geometries to unravel the role of inertia onthe stability of viscoelastic flow, however contradictory results were obtained. In Couette-Taylor

2469-990X/2018/3(10)/103302(10) 103302-1 ©2018 American Physical Society

ATUL VARSHNEY AND VICTOR STEINBERG

Nitrogen gas

45 mm

8 mm

1 mm 2.5 mm

WeighingBalance

28 mm

P

ΔP

21 mm

FIG. 1. Schematic of the experimental setup (not to scale). A differential pressure sensor, marked as �P ,is used to measure pressure drop across the obstacles. An absolute pressure sensor, marked as P , after thedownstream cylinder, is employed to obtain pressure fluctuations.

flow between two cylinders, the instability sets in at Witr, which grows with the elasticity numberEl (=Wi/Re) saturating at sufficiently high El and reduces with increasing inertia [22–24]. Incontrast, the onset of instability Retr is almost constant at very low El and decreases with increasingEl, in a rather limited range, in agreement with numerical simulations [25,26]. Recent experimentsin the Couette-Taylor flow with both inner and outer co- and counterrotating cylinders at lowEl show a weak smooth dependence on El [27–29]. At moderate El, either stabilization ordestabilization of the first bifurcation depending on co- or counterrotation of cylinders is found[27]. However, the general tendency in the dependences of the bifurcations on El reported inRefs. [22–24] was confirmed later in Refs. [27–29]. In contrast, a nonmonotonic dependence of thefirst bifurcation in a wall-bounded plane Poiseuille flow on El in its narrow range of low values wasrevealed in numerical simulations using the Oldroyd-B constitutive equation. The reduced solventviscosity strongly modifies this effect: The smaller the polymer contribution to the viscosity, the lesspronounced the effect [30]. In extensional viscoelastic flow [31], e.g., planar flow with an abruptcontraction and expansion, and in flow past a cylinder [32], the role of both elasticity and inertiawas investigated in a narrow range of Re and Wi and for only three El values. In extensional flow,the onset of the elastic instability Witr turns out to be independent of Re in the range of 0.1–40 forthree polymer solutions that correspond to El = 3.8, 8.4, and 89. However, in the case of flow past acylinder [32] Witr decreases with increasing Re. Recent numerical studies [33] on two-dimensionalviscoelastic flow past a cylinder reveal the phase diagram in (Wi, Re) coordinates and both dragenhancement and drag reduction (DR) were observed in the ranges Wi ≈ 0.2–10 and Re ≈ 0.1–105.Thus, despite extensive theoretical and experimental efforts, the influence of inertia on viscoelasticflow in a broad range of (Re, Wi) and El is still not understood and a stability diagram of differentflow regimes is lacking.

Here we perform experiments, over a broad range of (Re, Wi) and El, in a channel flow of dilutepolymer solution hindered by two widely spaced obstacles (see Fig. 1 for the experimental setup).Changing the solvent viscosity ηs by two orders of magnitude allows us to vary the elasticity numberEl = λ(ηs )ηs/ρD2 ∼ η2

s /ρD2 [34] by more than four orders of magnitude. Such an approachenables us to investigate the role of inertia in viscoelastic flow in different flow regimes in a widerange of (Re, Wi) and El.

The main feature of viscoelastic flow at Re � 1 between two widely spaced obstacles is anelastic wake instability in the form of quasi-two-dimensional counterrotating elongated vorticesgenerated by a reversed flow [35]. The two vortices constitute two mixing layers with a nonuniformshear velocity profile filling the interobstacle space. A further increase of Wi leads to chaoticdynamics with properties similar to ET [11]. There are several reasons for the choice of the flowgeometry. (i) Since the blockage ratio D/w � 1, the flow between the cylinders is unbounded, like“an island in a sea” of otherwise laminar channel flow, contrary to all previous wall-dominated flowgeometries that were used to study ET (D and w are the cylinders’ diameter and channel width,respectively) [3,5–9]. Therefore, we expect to observe mostly homogeneous, though anisotropic,flow closer to that considered in theory [12,13] and numerical simulations [14–16]. By employing

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DRAG ENHANCEMENT AND DRAG REDUCTION IN …

unbounded flow, we concentrate on variation in the bulk flow structures due to polymer additives,which results in a significant frictional loss. (ii) Large Wi and Re can be reached in the samesystem to scan the range from ET to DR. (iii) Several techniques can be simultaneously employedto quantitatively characterize the flow.

II. EXPERIMENT

A. Experimental setup and materials

The experiments are conducted in a linear channel of L × w × h = 45 × 2.5 × 1 mm3, shownschematically in Fig. 1. The fluid flow is hindered by two cylindrical obstacles of diameter D = 0.30mm made of stainless steel separated by a distance of e = 1 mm and embedded at the center of thechannel. Thus the geometrical parameters of the device are D/w = 0.12, h/w = 0.4, and e/D =3.3. The channel is made from transparent acrylic glass [poly(methyl methacrylate)]. The fluid isdriven by N2 gas at a pressure up to ∼60 psi and injected via the inlet into a rectangular channel. Asa fluid, a dilute polymer solution of high-molecular-weight polyacrylamide [PAAm, homopolymerof molecular weight Mw = 18 MDa (Polysciences)] at a concentration c = 80 ppm (c/c∗ � 0.4,where c∗ = 200 ppm is the overlap concentration for the polymer used [34]) is prepared using water-sucrose solvent with a sucrose weight fraction varying from 0 to 60% (see Table 1 in [36]). Thesolvent viscosity ηs at 20 ◦C is measured in a commercial rheometer (AR-1000, TA Instruments).An addition of polymer to the solvent increases the solution viscosity η of about 30%. The stress-relaxation method [34] is employed to obtain λ; for ηs = 0.1 Pa s solution, λ is measured to be10 ± 0.5 s. A linear dependence of λ on η was shown in Ref. [34].

B. Pressure measurements and imaging system

High-sensitivity differential pressure sensors (HSC series, Honeywell) of different ranges areused to measure the pressure drop �P across the obstacles and an additional absolute pressuresensor (ABP series, Honeywell) of different ranges is used to measure the pressure P fluctuationsafter the downstream cylinder at a sampling rate of 200 Hz, as shown schematically in Fig. 1. Theaccuracy of the pressure sensors used is ±0.25% full scale. We measure pressure drop for bothsolvent and polymer solution as a function of flow speed, and the difference between these twomeasurements provides information about the influence of polymers on the frictional drag.

The fluid exiting the channel outlet is weighed instantaneously W (t ) as a function of time t

by a PC-interfaced balance (BA210S, Sartorius) with a sampling rate of 5 Hz and a resolution of0.1 mg. The time-averaged fluid discharge rate Q̄ is estimated as �W/�t . Thus the flow speed iscalculated as U = Q̄/ρwh. For flow visualization, the solution is seeded with fluorescent particlesof diameter 1 μm (Fluoresbrite YG, Polysciences). The region between the obstacles is imagedin the midplane via a microscope (Olympus IX70), illuminated uniformly with light-emittingdiode (Luxeon Rebel) at 447.5-nm wavelength and two CCD cameras attached to the microscope,(i) GX1920 Prosilica with a spatial resolution of 1936 × 1456 pixels at a rate of 50 frames/s (fps)and (ii) a high-resolution CCD camera XIMEA MC124CG with a spatial resolution of 4112 × 3008pixels at a rate of 1 fps, are used to record the particles’ streaks.

III. RESULTS

Frictional drag f for each El is calculated through the measurement of pressure drop acrossthe obstacles �P (see Fig. 1) as a function of U and is defined as f = 2Dh�P/ρU 2Lc; Dh =2wh/(w + h) = 1.43 mm is the hydraulic radius and Lc = 28 mm is the distance between locationsof �P measurement [35]. Figure 2 shows variation of f with Re for three El values and a sequenceof transitions can be identified for each El. These transitions are further illustrated through a high-resolution plot of the normalized friction factor f/flam versus Re and Wi presented in the top andbottom insets of Fig. 2, respectively. Three flow regimes characterized by different scaling exponents

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ATUL VARSHNEY AND VICTOR STEINBERG

0.01 0.1 1 10 100 1000

0.01 0.1 1 10 100 1000

Re

Re

0.1

1

10

100

1000

f

1

2

3

45

f/flam

10 100 1000 100001

2

3

45

Wi

El=1.4El=149El=2433

0.5

0.2

0.5

0.2

f/flam

flam

FIG. 2. Friction factor f versus Re for three values of El. The dashed line flam ∼ 1/Re represents thelaminar flow. The top inset shows the normalized friction factor f/flam versus Re. The bottom inset shows thesame data presented as f/flam versus Wi with the fits marked by dashed lines in two regions: f/flam ∼ Wi0.5

above the elastic instability and f/flam ∼ Wi0.2 in the ET regime. Note that the drag reduction for El = 2433occurs at Re ≈ 0.5 and Wi ≈ 1216 and continues until the flow relaminarizes.

are identified. (i) The first drag enhancement above the elastic instability follows f/flam ∼ Wi0.5 forall values of El explored; for high El it is associated with a growth of two elongated vortices (or twomixing layers) [35]. (ii) Further drag enhancement at high El occurs due to ET [11] characterizedby a steep algebraic decay both in the power spectra of velocity and pressure fluctuations withexponents greater than ∼3 (discussed in the following) and in intensive vorticity dynamics anda growth of average vorticity as ω̄ ∼ Wi0.2 and f/flam ∼ Wi0.2, typical for ET [11]. For low El,either a saturation or a reduction of the friction factor with Re or Wi marks the DR regime. (iii)For both high and intermediate El, the DR regime with decreasing f/flam at increasing Re or Wiis observed and at low El the drag enhancement is noticed. Another striking finding is a completerelaminarization of flow, i.e., 100% drag reduction, that occurs for El = 2433 (also for El = 1070and 3704; data not shown), where f/flam returns to the laminar value at Re ≈ 4 (Wi ≈ 104). Withdecreasing El, the transition points are shifted to a higher value of Re and Wi, and remarkably evenat Re � 1 both drag enhancement and DR regimes can be recognized.

To elucidate further, the critical values of the respective transitions for each El is mapped inRe-El [Fig. 3(a)], Wi-El [Fig. 3(b)], and Wi-Re [Fig. 3(c)] coordinates. In the range explored for(Re, Wi), three different transitions are observed, which are associated with elastic instability, dragenhancement, and DR as shown in Figs. 3(a) and 3(b). These transitions persist for all elasticityvalues and the elastic instability occurs first, followed by the other two transitions. In addition,the complete flow relaminarization is observed only for El = 1070, 2433, and 3704. Interestingly,the sequence of DR and drag enhancement changes as El varies from low to high values; DRis followed by drag enhancement at low El and this sequence reverses at high El, as describedabove. This change in the sequence occurs in the intermediate range of elasticity at El ∼ 149.Furthermore, three regions in Figs. 3(a)–3(c) can be identified based on variation of the criticalvalues (Retr, Witr) with El. For low elasticity (El � 20), Retr is independent of El, while for highelasticity (El � 300), Retr drops sharply with El. For intermediate elasticity (20 � El � 300), Retr

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DRAG ENHANCEMENT AND DRAG REDUCTION IN …

(a)

0.001

0.01

0.1

1

10

100

1000

1 10 100 1000 10000

Re

El10

100

1000

10000

1 10 100 1000 10000W

iEl

(b)

0.001 0.01 0.1 1 10 100 100010

100

1000

10000 (c)

Wi

Re

El=1

4803 11

232

3704

2433

1070

304

149

62.3

31

10 6.53

1.4

FIG. 3. Stability diagram of different flow regimes in (a) Re-El, (b) Wi-El, and (c) Wi-Re coordinates.The colors of the symbols signify different transitions: black, first elastic instability; green, DR; blue, dragenhancement; and red, flow relaminarization. The gray band in (a)–(c) indicates the region of intermediate El.Solid lines of different colors in (c) are used as a guide to the eye to track the transition in different regions.

shows a weak dependence on El [see Fig. 3(a)]. In Fig. 3(b) the dependence of Witr on El isnonmonotonic: a strong growth with El at low El, a sharp decrease at high El, and a gradualgrowth at intermediate El. The transitions are further mapped in the Wi-Re plane for different El

to emphasize the role of inertia on the stability of a viscoelastic fluid flow. The same three regionsare identified: At high El, Witr grows with Retr with a stabilizing effect of inertia, at low El thereis a steep drop of Witr with Retr, and in the intermediate region Witr decreases with increasing Retr

with the destabilizing effect of inertia [see Fig. 3(c)].Long-exposure particle streak images in Fig. 4 illustrate the flow structures in three regions of

elasticity and at different Re and Wi above the transitions’ values (see also the corresponding moviesSM1–SM9 in [36]). In low- and intermediate-elasticity regions, a large-scale vortical motion appearsabove the elastic instability; however, in DR and drag enhancement regimes, small-scale turbulentstructures dominate and the large-scale vortical motion vanishes (top and middle panels in Fig. 4).In a high-elasticity region, e.g., El = 14 803, an unsteady pair of vortices [35] spans the regionbetween the obstacles (e.g., Re = 0.005 and Wi = 74) and at higher-Re small-scale vortices emergewith an intermittent and random dynamics (e.g., Re = 0.03 and Wi = 444) that constitutes the ET

Re=303.4, El=10

Re=4, El=62.3 Re=10, El=62.3 Re=30, El=62.3

Re=0.005, El=14803 Re=0.03, El=14803 Re=0.57, El=14803

Re=54.3, El=10 Re=153.4, El=10

FIG. 4. Representative snapshots of flow structures in the three regions, above the transitions, at El = 10(top panel), El = 62.3 (middle panel), and El = 14 803 (bottom panel); see also the corresponding moviesSM1–SM9 in [36]. The scale bars are 100 μm.

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ATUL VARSHNEY AND VICTOR STEINBERG

El=1.4S(

P) (a

.u.)

El=149

S(P)

(a.u

.)

El=14803

S(P)

(a.u

.)

(a) (b) (c)

λν λνλν0.01 0.1 1

102

103

104

Re=367.7, Wi=531.2Re=613.5, Wi=886.1Re=646.1, Wi=933.3Re=813.7, Wi=1175.3Re=844.9, Wi=1220.4Re=903.1, Wi=1304.4

-3

-1.3

0.01 0.1 1

102

103

104

105

-1.3

Re=4.07, Wi=608.2

Re=5.7, Wi=854.2Re=7.3, Wi=1095.6

Re=5.3, Wi=789.4

Re=8.9, Wi=1334.7

Re=4.7, Wi=699

0.1 1 10 100

102

103

104

105

106

-3

Re=0.014, Wi=203.6

Re=0.07, Wi=1030.1Re=0.083, Wi=1228.3

Re=0.062, Wi=921.8

Re=0.11, Wi=1564.1

Re=0.056, Wi=837.5

Re=0.12, Wi=1827.9

FIG. 5. Pressure power spectra S(P ) versus normalized frequency λν in the drag enhancement regime atvarious Re and Wi and in three regions of elasticity (a) El = 1.4, (b) El = 149, and (c) El = 14 803. Thedashed line shows power-law decay with an exponent β specified beside the line.

regime [11], whereas in the DR regime (e.g., Re = 0.57 and Wi = 8438) a much smoother spatialscale and less vortical motion are found (bottom panel of Fig. 4). However, a quantitative analysisof the velocity field at low and intermediate values of El requires serious technical effort and isbeyond the scope of the present investigation.

Finally, we characterize the observed flow regimes through frequency power spectra of absolutepressure fluctuations for various Re and Wi in three regions of elasticity. The pressure spectra arepresented as a function of normalized frequency λν to signify the timescales involved in flow withrespect to λ. Figure 5 shows pressure power spectra S(P ) in the drag enhancement regime for threeEl values. For low elasticity, the S(P ) decay exponent β evolves from −1.3 to −3 with increasingRe and Wi and in the range of λν ∼ 0.2–1 [shown in Fig. 5(a) for El = 1.4]. It is worth noting thatβ ≈ −3 is reached at the highest Re and Wi. In the intermediate range of elasticity, the exponentvalue β = −1.3 is obtained in the range of λν ∼ 0.1–1, the same as for low El [shown in Fig. 5(b)for El = 149], whereas for high El, S(P ) exhibits steep decay with β ≈ −3 in a higher-frequencyrange λν ∼ 1–10 for all Re and Wi values for El = 14 803 [Fig. 5(c)]. This value of β is oneof the main characteristics of the ET regime [8,11]. The value λν = 1 at high El is a relevantfrequency to generate ET spectra with β ≈ −3 at higher frequencies [8], as the stretching-and-folding mechanism of elastic stresses due to the velocity field redistributes energy across the scales[12,13]. Similar scaling of S(P ) is observed in numerical simulations in the dissipation range ofthe turbulent drag reduction regime [17]. For low El, the S(P ) decay at λν > 0.1 up to 1 is causedby the inertial effect. In the drag reduction regime, S(P ) demonstrates completely different scalingbehavior with λν, shown in Fig. 6. For low El, one finds a steep decay of S(P ) at high frequenciesλν � 1 with a scaling exponent ∼−3.4 and a rather slow decay with an exponent between −0.5 and−1 at λν < 1 [Fig. 6(a) for El = 1.4], in accord with numerical simulations [17]. For high El, thespectra S(P ) decay steeply at high frequency λν ∼ 10, and at low frequencies 0.1 < λν ∼ 10 a slowdecay with an exponent ∼−1 is observed [shown in Fig. 6(c) for El = 14 803]. In the intermediaterange of El, the decay exponent varies between −1.8 and −2.5 [shown in Fig. 6(b) for El = 149] inthe frequency range 0.1 < λν ∼ 10. To highlight the scaling dependences of S(P ) between differentflow regimes in each El region, we present the same data in Fig. S1 in [36] for various Re and Wiand for three regions of El.

For comparison we present f as well as f/flam as a function of Re in the range between ∼6and ∼900 for two Newtonian fluids, water (ηs = 1 mPa s) and a solution of 25% sucrose in water

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DRAG ENHANCEMENT AND DRAG REDUCTION IN …

El=1.4

S(P)

(a.u

.)

El=149

S(P)

(a.u

.)

El=14803

S(P)

(a.u

.)

(a) (b) (c)

λν λν λν

-3.4

-1

0.01 0.1 1

102

103

104

10

Re=142.9, Wi=208.6Re=171.4, Wi=250.3Re=193.3, Wi=282.2Re=222.2, Wi=324.4

-1

Re=0.43, Wi=6417.2

Re=0.186, Wi=2760.4Re=0.23, Wi=3433.5Re=0.34, Wi=5026.2

Re=0.48, Wi=7086.2Re=0.57, Wi=8485.7

0.1 1 10 10010

102

103

104

105

106

0.1 1 10

102

103

104

Re=28.3, Wi=4219.7

Re=11.9, Wi=1786.1Re=14.9, Wi=2232.1Re=25.8, Wi=3860.8

Re=30.4, Wi=4546.5

-1.8

-2.5

FIG. 6. Pressure power spectra S(P ) versus λν in the DR regime at various Re and Wi and in three regionsof elasticity (a) El = 1.4, (b) El = 149, and (c) El = 14 803. The dashed line shows power-law decay with anexponent specified beside the line.

(ηs = 3 mPa s) (see Fig. S2 in [36]). The dependences of both f and f/flam on Re are smooth andgrowing at Re � 70, which differs significantly from that found for polymer solutions at low El. Itis also rather different from the dependence of f on Re in a channel flow past an obstacle, whichis studied extensively. Thus, we can conclude that in viscoelastic flow, we observe inertia-modifiedelastic instabilities, contrary to inertial instabilities modified by elastic stress. This conclusion isfurther supported by the measurements of the power spectra of pressure fluctuations in a Newtonianfluid flow that exhibit a power-law exponent ∼−1.6 at high Re (see Fig. S3 in [36]), in contrast tothose presented in Figs. 5 and 6.

IV. DISCUSSION AND CONCLUSION

Polymer degradation is often encountered under strong shear and in particular at high elongationrates due to velocity fluctuations at Re � 1 and Wi � 1 [37]. As a result of degradation, theinfluence of polymers on the flow becomes ineffective. To ensure that the drag enhancement andDR we observe in our experiments at Re � 1 are not the result of polymer degradation, we reusethe polymer solutions (after performing the experiment with two obstacles) in experiments on achannel flow with a single obstacle. Indeed, we observe elastic instability, drag enhancement, andDR with a single obstacle for El = 14 803 (see, e.g., Fig. S4 in [36]). Moreover, the problemof polymer degradation was addressed in detail in our paper on turbulent drag reduction in alarge-scale swirling flow experiment conducted at Re � 2 × 106 [38]. It was pointed out that “themain technical achievement in the experiment was long term stability of polymers in turbulent flowthat allowed us to take large data sets up to 106 data points for up to 3.5 hours at the highest Rewithout a sign of polymer degradation.” Thus, we conclude that the observed flow regimes in ourexperiments are not caused by polymer degradation.

The presented results on the friction coefficient and the pressure power spectra obtained in awide range of controlled parameters exhibit two remarkable features: (i) the presence of three flowregimes with distinctive and different scaling behavior in both f/flam and S(P ) and (ii) three regionson the stability diagrams in the planes of Re, Wi, and El parameters depending on the value of fluidelasticity. In spite of the fact that rather high values of Re are reached, inertial turbulence is notattained in the region between the obstacles and channel flow outside this region. As known fromthe literature, turbulence in a flow past an obstacle is attained at much higher Re [39].

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The different scaling dependences of S(P ) in three flow regimes and in three regions of elasticityindicate the intricate interaction between elastic and inertial stresses. A two-way energy transferbetween turbulent kinetic energy and elastic energy of polymers also results in a modification ofthe velocity spectra’s scaling exponents at Re � 1 [20,21]. The effect of inertia at Re ∼ 100 onthe scaling behavior of velocity power spectra with the exponent |α| ≈ 2.2 instead of ∼3.5 in pureET was observed experimentally in the Couette-Taylor viscoelastic flow [6] and later confirmednumerically [40]. What is remarkable is that in the drag enhancement regime about the same scalingexponent β ≈ −3 in S(P ) is found for low and high El at close values of Wi and a three order ofmagnitude difference in Re values. This indicates the elastic nature of drag enhancement regimesat both low and high El. Indeed, the scaling exponents of the pressure power spectrum decay forEl = 1.4 [Fig. 5(a)] show |β| ∼= 3 at Re >∼ 845 and Wi >∼ 1220. The observations of scalingf/flam ∼ Wi0.2, the exponent of the pressure spectrum decay |β| ∼ 3, and the exponent of thevelocity spectrum decay |α| ∼ 3.5 are characteristics of ET flow [11]. Thus, the drag enhancementregime in low-El regions is typical of ET.

A striking and unanticipated observation in the high-elasticity region is a significant DR and acomplete flow relaminarization at Wi > 1000 and Re ∼ O(1) (see Figs. 2 and 3). The obtainedresult is different from turbulent DR observed at Re � 1, where Reynolds stress exceeds the elasticstress prior to the onset of turbulent DR and becomes comparable to elastic stress at the onset.A similar effect of the saturation and even weak reduction of f/flam was observed and discussedin the planar geometry with an abrupt contraction and expansion of a microfluidic channel flow,where the saturation of f/flam at higher polymer concentrations c and even its reduction at lowerc < c∗ were revealed in the range 0 < Wi < 500 for three polymer solutions of different polymerconcentrations [31]. For the highest c, f/flam reached a value of ∼3.5 at high Wi, in agreement withthe early measurements in a pipe flow with an axisymmetric contraction and expansion at muchlower Wi < 8 [41].

To find a possible explanation of DR in a wide range of El and (Re, Wi), we discuss the effectin details. At low El between 1.4 and 31, either drag saturation or weak DR occurs just before thedrag enhancement regime associated with ET and discussed above. It is worth mentioning that, dueto the intricate interplay between elastic and inertial stresses, the strength of DR is a nonmonotonicfunction of El and depends on the relation between Wi and Re. The higher the Wi and the lowerthe Re, the more pronounced the DR regime at low El. The range of Re observed in the DR regimecorresponds to the vorticity suppression by elastic stress generated by polymer additives injectedinto a Newtonian fluid flow [42–44], which is indeed confirmed by the snapshots at El = 10 andRe = 54 and 153, shown in Fig. 4.

At high El in the range 149 < El < 14 803 and Re <∼ 70, f/flam reduces significantly.However, the complete flow relaminarization is observed only at El = 1070, 2433, and 3704,where Wi � 104 and Re < 10. This means that high values of Wi and Re stabilize DR prior to therelaminarization due to finite polymer extensibility. The snapshot at El = 14 803 and Re = 0.57 inFig. 4 shows a vorticity-free flow, in contrast to the snapshots at the same El and low Re. Thus,at Re ∼ O(1) and high Wi, the inertial effects are negligible to suppress the growth of f/flam,whereas at Re � 70 and intermediate values of Wi, the drag can be saturated, as seen, for example,at El = 31. As suggested in Ref. [31], the saturation of f/flam observed for high El is probably aconsequence of polymer chains attaining their finite-extensibility limit at very-high-Wi values. Herewe emphasize again that the DR observed at low El and high Re is not related to the turbulent dragreduction realized in a turbulent flow at higher values of Re than achieved in our experiment.

The theory of ET and the corresponding numerical simulations do not consider the inertial effectsand their role in ET and therefore they are unable to explain the DR and flow relaminarization. Thus,the results reported call for further theoretical and numerical development to uncover inertial effectson viscoelastic flow in a broad range of (Re, Wi) and El.

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ACKNOWLEDGMENTS

We thank Guy Han and Yuri Burnishev for technical support. We are grateful to Dr. DongyangLi for his help in the measurements of the friction factor and the pressure spectra of Newtonian fluidflow. A.V. acknowledges support from the European Union’s Horizon 2020 research and innovationprogram under the Marie Skłodowska-Curie Grant Agreement No. 754411. This work was partiallysupported by Israel Science Foundation (Grant No. 882/15) and Binational U.S.-Israel Foundation(Grant No. 2016145).

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